Repdigits in k-generalized Pell sequence

Abstract

Let $k\ge 2$ and let $(P_n^{(k)}{)}_{n\ge 2-k}$ be $k$-generalized Pell sequence defined by \begin{equation*}P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+...+P_{n-k}^{(k)}\end{equation*} for $n\ge 2$ with initial conditions \begin{equation*}P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end{equation*} In this paper, we deal with the Diophantine equation \begin{equation*}P_{n}^{(k)}=d\left( \frac{10^{m}-1}{9}\right)\end{equation*} in positive integers $n,m,k,d$ with $k\ge 2,$ $m\ge 2$ and $1\le d\le 9$. We will show that repdigits with at least two digits in the sequence ${\left(P_n^{(k)}\right)}_{n\ge 2-k}$ are the numbers\ $P_5^{(3)}=33$ and $P_6^{(4)}=88.$